How can "any process you can imagine" be thought of as function computation?
A function is simply a procedure that maps a particular input to a particular output. You put in $X$, and the function computes $Y$. Those $X$ and $Y$ can take many different forms. It could be mapping one number to another number (convert miles to kilometres), mapping sound to text (name that tune), mapping text to text (translate languages), mapping a video to text (review this movie), or mapping text to an image (draw a picture of $X$). Anytime you have a procedure that produces a fixed output based on a fixed input, it's a function.
Universality theorems guarantee that a neural network can produce an arbitrarily good approximation of any possible function. That doesn't mean it's easy, though - finding the right function that maps $X$ to $Y$ is the hard part.